Space of modular forms of level 3600, weight 1, and Character 3600.e

• Label: 3600.1.e
• Level: $3600$
• Weight: $1$
• Character orbit: 3600.e
• Rep. character: $\chi_{3600}(3151,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $720$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3600.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(720\)
Trace bound:			\(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(3600, [\chi])\).

	Total	New	Old
Modular forms	90	6	84
Cusp forms	18	6	12
Eisenstein series	72	0	72

The following table gives the dimensions of subspaces with specified projective image type.

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	6	0	0	0

Trace form

\( 6 q + 2 q^{13} + 2 q^{29} - 2 q^{37} + 2 q^{41} - 6 q^{49} - 4 q^{61} + 2 q^{73} + 2 q^{89} + 2 q^{97}+O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(3600, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3600.1.e.a	$3600$	$1$	3600.e	4.b	$2$	$1$	$1$	$1.797$	\(\Q\)	$D_{2}$	\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \)	\(\Q(\sqrt{5}) \)	✓				80.1.h.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q+2q^{29}+2q^{41}+q^{49}-2q^{61}+2q^{89}+\cdots\)
3600.1.e.b	$3600$	$1$	3600.e	4.b	$2$	$1$	$1$	$1.797$	\(\Q\)	$D_{2}$	\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{3}) \)	✓				144.1.g.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q+2q^{13}-2q^{37}+q^{49}+2q^{61}+2q^{73}+\cdots\)
3600.1.e.c	$3600$	$1$	3600.e	4.b	$2$	$2$	$2$	$1.797$	\(\Q(\sqrt{-3}) \)	$D_{6}$	\(\Q(\sqrt{-3}) \)	None		✓			3600.1.e.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$		\(q+(\zeta_{6}+\zeta_{6}^{2})q^{7}-q^{13}+(\zeta_{6}+\zeta_{6}^{2}+\cdots)q^{19}+\cdots\)
3600.1.e.d	$3600$	$1$	3600.e	4.b	$2$	$2$	$2$	$1.797$	\(\Q(\sqrt{-3}) \)	$D_{6}$	\(\Q(\sqrt{-3}) \)	None		✓			3600.1.e.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$		\(q+(-\zeta_{6}-\zeta_{6}^{2})q^{7}+q^{13}+(\zeta_{6}+\zeta_{6}^{2}+\cdots)q^{19}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(3600, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(3600, [\chi]) \simeq \) \(S_{1}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 3}\)